Polynomials vanishing on Cartesian products: The Elekes-Szabó Theorem revisited

Abstract

Let $F\in \mathbb{C}[x,y,z]$ be a constant-degree polynomial,and let $A,B,C\subset \mathbb{C}$ be finite sets of size $n$. We show that $F$ vanishes on at most $O(n^{11/6})$ points of the Cartesian product $A\times B\times C$, unless $F$ has a special group-related form. This improves a theorem of Elekes and Szabó [Combinatorica, 2012], and generalizes a result of Raz, Sharir, and Solymosi [Amer. J. Math., to appear]. The same statement holds over $\mathbb{R}$, and a similar statement holds when $A,B,C$ have different sizes (with a more involved bound replacing $O(n^{11/6})$). This result provides a unified tool for improving bounds in various Erd\H os-type problems in combinatorial geometry, and we discuss several applications of this kind.

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Cited by in corpus (13)

• Point-curve incidences in the complex plane
• Projective geometries arising from Elekes-Szabó problems
• On sets defining few ordinary circles
• Improved Elekes-Szabó type estimates using proximity
• Schwartz-Zippel bounds for two-dimensional products
• The number of unit-area triangles in the plane: Theme and variations
• On the dimension of exceptional parameters for nonlinear projections, and the discretized Elekes-Rónyai theorem
• A note on distinct distances
• Convexity, Squeezing, and the Elekes-Szabó Theorem
• On triple intersections of three families of unit circles
• Dimension-expanding polynomials and the discretized Elekes-Rónyai theorem
• Model-theoretic Elekes-Szabó in the strongly minimal case
• Convex Polygons in Cartesian Products